In Bogachev's book "Gaussian Measures" (Example 3.8.13) sufficient conditions for the (uniform) tightness of a sequence of centered Borel Gaussian probability measures on a separable Hilbert spaces in terms of the covariance operators are given. In particular, a sequence of such measures is tight in the weak topology if the covariance operators are uniformly bounded, and is tight in the norm topology if the traces of the covariance operators are uniformly bounded and the series defining the traces converge uniformly.
My question is: Do similar criteria for uniform tightness (in terms of covariance operators) exist for centered Gaussian measures on a reflexive separable Banach space? I'm interested in particular in Gaussian measures on $L^p$-spaces for which a characterization of the covariance operators is known (see 1).
